Bounding the Partition Function of Spin-Systems

With a graph G = (V,E) we associate a collection of non-negative real weights ⋃ v∈V {λi,v : 1 ≤ i ≤ m} ∪ ⋃ uv∈E{λij,uv : 1 ≤ i ≤ j ≤ m}. We consider the probability distribution on {f : V → {1, . . . ,m}} in which each f occurs with probability proportional to ∏ v∈V λf(v),v ∏ uv∈E λf(u)f(v),uv . Many well-known statistical physics models, including the Ising model with an external field and the hard-core model with non-uniform activities, can be framed as such a distribution. We obtain an upper bound, independent of G, for the partition function (the normalizing constant which turns the assignment of weights on {f : V → {1, . . . ,m}} into a probability distribution) in the case when G is a regular bipartite graph. This generalizes a bound obtained by Galvin and Tetali who considered the simpler weight collection {λi : 1 ≤ i ≤ m} ∪ {λij : 1 ≤ i ≤ j ≤ m} with each λij either 0 or 1 and with each f chosen with probability proportional to ∏ v∈V λf(v) ∏ uv∈E λf(u)f(v). Our main tools are a generalization to list homomorphisms of a result of Galvin and Tetali on graph homomorphisms and a straightforward secondmoment computation.